Since the inverse scattering transform (IST) technique, one of the most powerful tool to study solitons of nonlinear PDEs, was firstly
presented to solve the initial-value problem for the KdV equation by Gardner, Greene, Kruskal and Miurra [1]},
there have been some results on the mKdV equation. For instance, Wadati studied the focusing mKdV equation with zero boundary conditions
and obtained simple-pole, double-pole and triple-pole solutions [2]}, [3]}, after which the \(N\) -soliton solutions and breather
solutions for the focusing mKdV was derived by Demontis [4]}. Recently, Zhang and Yan presented the inverse scattering transform for the
focusing and defocusing mKdV equation for nonzero boundary conditions in terms of the matrix-valued Riemann-Hilbert (RH) problem [5]}.
